Let $\xi(t)$ be a Wiener process in $E_n$ , $\alpha_n$ a non-anticipative vector function, $\delta=\{\alpha_t\}$ , $x_t^{\delta,x}$ a solution of $$ x_t=x+\int_0^t\sigma(x_s,\alpha_s)d\xi_s+\int_0^t b(x_s,\alpha_s)\,ds, $$$\varphi=\varphi(x)$. In this paper, smouthness of functions$$ v(x)=\sup_{\delta,\tau}\mathbf{M}\biggl[\int_0^\tau e^{-\lambda t}f(x_t^{\delta,x},\alpha_t)\,dt+e^{-\lambda\tau}\varphi(x_\tau^\delta,x)\biggr] $$is investigated.Under conditions of smouthness type on$\sigma,b,f,\varphi$it is proved that$v\in W_{p,\textrm{loc}}^2$(Sobolev space). If, in addition,$\sigma\sigma^*$is strictly positive-definite, then$$ \sup_\alpha (L^\alpha v+f^\alpha)\leq 0\ (\textrm{a.e.}), \quad \sup_\alpha (L^\alpha v+f^\alpha)=0\ (\textrm{a.e.}\ \{x: v(x)>\varphi(x)\}). $$The structure of$\varepsilon$-optimal policies$\delta$and$\varepsilon$-optimal stopping times$\tau$is also studied.